We construct a family of subdistributions of the Levi core $\mathfrak{ C}(\mathcal{ N})$ called modified Levi cores $\{ \mathcal{ M}\mathfrak{ C}_{ \mathcal{ A}}\}_{ \mathcal{ A}}$ indexed over closed distributions $\mathcal{ A}$ that contain the Levi null distribution $\mathcal{ N}$ and are contained in the complex tangent bundle $T^{ 1, 0}b\Omega$ of a smooth bounded pseudoconvex domain $\Omega$. We show that Catlin's Property ($P$) holds on $b\Omega$ if and only if Property ($P$) holds on the support of one, and hence all, of the modified Levi cores. In $\mathbb{ C}^2$, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in $\mathbb{ C}^2$ that satisfies Property ($P$), we show that its modified Levi core is trivial. This contrasts with $\mathfrak{ C}(\mathcal{ N})$, which can be nontrivial for such domains.
Categories: math.CV